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Lattices can be considered either as special sorts of partially ordered sets, with a signature consisting of one binary relation symbol ≤, or as algebraic structures with a signature consisting of two binary operations ∧ and ∨. The two approaches can be related by defining a≤ b to mean a∧b=a. For two binary operations the axioms for a lattice are: Commutative laws: \forall a \forall b \; a \vee b = b \vee a \forall a \forall b\; a \wedge b = b \wedge a Associative laws: \forall a \forall b \forall c\; a \vee (b \vee c) = (a \vee b) \vee c \forall a \forall b \forall c\; a \wedge (b \wedge c) = (a \wedge b) \wedge c Absorption laws: \forall a \forall b \;a \vee (a \wedge b) = a \forall a \forall b \;a \wedge (a \vee b) = a For one relation ≤ the axioms are: Axioms stating ≤ is a partial order, as above. \forall a \forall b \exist c\; c\le a\wedge c\le b \wedge \forall d\;d\le a\wedge d\le b \rightarrow d\le c (existence of c=a∧b) \forall a \forall b \exist c\; a\le c\wedge b\le c \wedge \forall d\;a\le d\wedge b\le d \rightarrow c\le d (existence of c=a∨b) First order properties include: \forall x \forall y\forall z\;x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z) (distributive lattices) \forall x \forall y\forall z\;x \vee (y \wedge (x \vee z)) = (x \vee y) \wedge (x \vee z) (modular lattices) Completeness is not a first order property of lattice. |
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